The effect of a time delay on the transmission on dengue fever is studied. The time delay is due to the presence of an incubation period during which the virus replicates enough in the mosquito so that the viruses can be transmitted by the mosquito to a human. The conditions for the existence of a Hopf bifurcation to limit cycle behavior are established. A theorem for determining whether for a given set of parameter values which satisfies the mathematical conditions, the system will actually undergo a transition from a stable state into a limit cycle state is established. It is found that for a set of realistic values of the parameters in the model, a Hopf bifurcation will not occur even when it is mathematically possible. For a set of unrealistic values of some of the parameters, it is shown that a Hopf bifurcation can occur. Numerical solutions using this set of values show the trajectory of two of the variables making a transition from a spiraling orbit to a limit cycle orbit.